Journal of Integer Sequences, Vol. 25 (2022), Article 22.4.8 |

Department of Mathematics

Ludwig-Maximilians-Universität München

Theresienstraße 39

80333 Munich

Germany

Paul K. Stockmeyer

Department of Computer Science

The College of William & Mary

P. O. Box 8795

Williamsburg, VA 23187-8795

USA

**Abstract:**

Sierpiński graphs and Sierpiński triangle graphs
form two-parametric families of connected simple graphs which are related, for , to the Tower of Hanoi with discs and for
to the Sierpiński triangle fractal. The vertices of minimal degree play a special role as extreme vertices in and primitive vertices in
. The key concept of this note is that of an -key vertex whose distance to one of the extreme or primitive vertices, respectively, is times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter like, e.g., the Fibonacci sequence (golden) for and the Pell sequence (silver) for . The elements of most of these sequences form self-generating sets. We discuss the cases in detail.

(Concerned with sequences A000045 A000129 A000225 A000975 A001045 A002450 A002620 A003754 A004526 A005578 A006498 A023758 A048654 A052499 A070550 A089928 A089931 A097083 A181666 A182512 A247648 A353578 A353579 A353580 A353581 A353582.)

Received February 9 2022; revised versions received May 13 2022; June 9 2022.
Published in *Journal of Integer Sequences*,
June 10 2022.

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